Searching for Solutions

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Searching for Solutions

• Artificial Intelligence
• CMSC 25000
• January 15, 2008

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Agenda

• Search Motivation
• Problem-solving agents
• Rigorous problem definitions
• Blind exhaustive search
• Breadth-first search
• Uniform search
• Depth-first search
• Iterative deepening search
• Search analysis
• Computational cost, limitations

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Problem-Solving Agents

• Goal-based agents
• Identify goal, sequence of actions that satisfy
• Goal set of satisfying world states
• Precise specification of what to achieve
• Problem formulation
• Identify states and actions to consider in
  achieving goal
• Given a set of actions, consider sequence of
  actions leading to a state with some value
• Search Process of looking for sequence
• Problem -gt action sequence solution

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Agent Environment Specification

• Dimensions
• Fully observable vs partially observable Fully
• Deterministic vs stochastic Deterministic
• Static vs dynamic Static
• Discrete vs continuous Discrete
• Issues?

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Closer to Reality

• Sensorless agents (conformant problems)
• Replace state with belief state
• Multiple physical states, successorssets of
  successors
• Partial observability (contigency problems)
• Solution is tree, branch chosen based on percepts

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Example vacuum world

• Single-state, start in 5. Solution?

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Example vacuum world

• Single-state, start in 5. Solution? Right,
  Vacuum
• Sensorless, start in 1,2,3,4,5,6,7,8 e.g.,
  Right goes to 2,4,6,8 Solution?

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Example vacuum world

• Sensorless, start in 1,2,3,4,5,6,7,8 e.g.,
  Right goes to 2,4,6,8 Solution?
  Right,Vacuum,Left,Vacuum
• Contingency
• Nondeterministic Vacuum may dirty a clean
  carpet
• Partially observable location, dirt at current
  location.
• Percept L, Clean, i.e., start in 5 or
  7Solution?

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Example vacuum world

• Sensorless, start in 1,2,3,4,5,6,7,8 e.g.,
  Right goes to 2,4,6,8 Solution?
  Right,Vacuum,Left,Vacuum
• Contingency
• Nondeterministic Vacuum may dirty a clean
  carpet
• Partially observable location, dirt at current
  location.
• Percept L, Clean, i.e., start in 5 or
  7Solution? Right, if dirt then Vacuum

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Formal Problem Definitions

• Key components
• Initial state
• E.g. First location
• Available actions
• Successor function reachable states
• Goal test
• Conditions for goal satisfaction
• Path cost
• Cost of sequence from initial state to reachable
  state
• Solution Path from initial state to goal
• Optimal if lowest cost

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Selecting a state space

• Real world is absurdly complex
• ? state space must be abstracted for problem
  solving
• (Abstract) state set of real states
• (Abstract) action complex combination of real
  actions
• e.g., "Arad ? Zerind" represents a complex set of
  possible routes, detours, rest stops, etc.
• For guaranteed realizability, any real state "in
  Arad must get to some real state "in Zerind"
• (Abstract) solution
• set of real paths that are solutions in the real
  world
• Each abstract action should be "easier" than the
  original problem

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Why Search?

• Not just city route search
• Many AI problems can be posed as search
• Planning
• Vertices World states Edges Actions
• Game-playing
• Vertices Board configurations Edges Moves
• Speech Recognition
• Vertices Phonemes Edges Phone transitions

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Vacuum world state space graph

• states?
• actions?
• goal test?
• path cost?

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Vacuum world state space graph

• states? integer dirt and robot location
• actions? Left, Right, Vacuum
• goal test? no dirt at all locations
• path cost? 1 per action

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Example The 8-puzzle

• states?
• actions?
• goal test?
• path cost?

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Example The 8-puzzle

• states? locations of tiles
• actions? move blank left, right, up, down
• goal test? goal state (given)
• path cost? 1 per move
• Note optimal solution of n-Puzzle family is
  NP-hard

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Example robotic assembly

• states? real-valued coordinates of robot joint
  angles, parts of the object to be assembled
• actions? continuous motions of robot joints
• goal test? complete assembly
• path cost? time to execute

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Basic Search Algorithm

• Form a 1-element queue of 0 costroot node
• Until first path in queue ends at goal or no
  paths
• Remove 1st path from queue extend path one step
• Reject all paths with loops
• Add new paths to queue
• If goal foundgtsuccess else, failure

Paths to extend Order of paths added Position
new paths added
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Basic Search Problem

• Vertices Cities Edges Steps to next, distance
• Find route from S(tart) to G(oal)

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4
3
5
5
4
3
2
4
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Formal Statement

• Initial State in(S)
• Successor function
• Go to all neighboring nodes
• Goal state in(G)
• Path cost
• Sum of edge costs

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Blind Search

• Need SOME route from S to G
• Assume no information known
• Depth-first search, breadth-first search,
  uniform-cost search, iterative deepening search
• Convert search problem to search tree
• RootZero length path at Start
• NodePath label by terminal node
• Child one-step extension of parent path

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Search Tree
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Breadth-first Search

• Explore all paths to a given depth

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Breadth-first Search Algorithm

• Form a 1-element queue of 0 costroot node
• Until first path in queue ends at goal or no
  paths
• Remove 1st path from queue extend path one step
• Reject all paths with loops
• Add new paths to BACK of queue
• If goal foundgtsuccess else, failure

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Analyzing Search Algorithms

• Criteria
• Completeness Finds a solution if one exists
• Optimal Find the best (least cost) solution
• Time complexity Order of growth of running time
• Space complexity Order of growth of space needs
• BFS
• Complete yes Optimal only if steps cost
• Time complexity O(bd1) Space O(bd1)

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Uniform-cost Search

• BFS
• Extends path with fewest steps
• UCS
• Extends path with least cost
• Analysis
• Complete? Yes Optimal? Yes
• Time O(b(C/e)) Space O(b(C/e))

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Uniform-cost Search Algorithm

• Form a 1-element queue of 0 costroot node
• Until first path in queue ends at goal or no
  paths
• Remove 1st path from queue extend path one step
• Reject all paths with loops
• Add new paths to queue
• Sort paths in order of increasing length
• If goal foundgtsuccess else, failure

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Depth-first Search

• Pick a child of each node visited, go forward
• Ignore alternatives until exhaust path w/o goal

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Depth-first Search Algorithm

• Form a 1-element queue of 0 costroot node
• Until first path in queue ends at goal or no
  paths
• Remove 1st path from queue extend path one step
• Reject all paths with loops
• Add new paths to FRONT of queue
• If goal foundgtsuccess else, failure

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Search Issues

• Breadth-first search
• Good if many (effectively) infinite paths, bltlt
• Bad if many end at same short depth, bgtgt
• Uniform-cost search
• Tries paths with many short steps first
• Identical to BFS if all steps same cost
• Depth-first search
• Good if most partialgtcomplete, not too long
• Bad if many (effectively) infinite paths

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Iterative Deepening

• Problem
• DFS good space behavior
• Could go down blind path, or sub-optimal
• Solution
• Search at progressively greater depths
• 1,2,3,4,5..

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Progressive Deepening

• Question Arent we wasting a lot of work?
• E.g. cost of intermediate depths
• Answer (surprisingly) No!
• Most nodes at fringe
• Fringe (depth d) Cost bd
• Preceding depths b0 b1b(d-1)
• (bd - 1)/(b -1)
• Ratio of last ply cost/all preceding b - 1
• For large branching factors, prior work small
  relative to maximum depth

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Heuristic Search

• A little knowledge is a powerful thing
• Order choices to explore better options first
• More knowledge gt less search
• Better search alg?? Better search space
• Measure of remaining cost to goal-heuristic
• E.g. actual distance gt straight-line distance

A
B
C
10.4
6.7
4.0
11.0
S
G
3.0
8.9
6.9
D
E
F
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Hill-climbing Search

• Select child to expand that is closest to goal

S
8.9
A
D
10.4
6.9
E
10.4
A
B
F
3.0
6.7
G
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Hill-climbing Search Algorithm

• Form a 1-element queue of 0 costroot node
• Until first path in queue ends at goal or no
  paths
• Remove 1st path from queue extend path one step
• Reject all paths with loops
• Sort new paths by estimated distance to goal
• Add new paths to FRONT of queue
• If goal foundgtsuccess else, failure

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Beam Search

• Breadth-first search of fixed width - top w
• Guarantees limited branching factor, E.g. w2

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Beam Search Algorithm

• Form a 1-element queue of 0 costroot node
• Until first path in queue ends at goal or no
  paths
• Extend all paths one step
• Reject all paths with loops
• Sort all paths in queue by estimated distance to
  goal
• Put top w in queue
• If goal foundgtsuccess else, failure

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Best-first Search

• Expand best open node ANYWHERE in tree
• Form a 1-element queue of 0 costroot node
• Until first path in queue ends at goal or no
  paths
• Remove 1st path from queue extend path one step
• Reject all paths with loops
• Put in queue
• Sort all paths by estimated distance to goal
• If goal foundgtsuccess else, failure

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Heuristic Search Issues

• Parameter-oriented hill climbing
• Make one step adjustments to all parameters
• E.g. tuning brightness, contrast, r, g, b on TV
• Test effect on performance measure
• Problems
• Foothill problem aka local maximum
• All one-step changes - worse!, but not global max
• Plateau problem one-step changes, no FOM
• Ridge problem all one-steps down, but not even
  local max
• Solution (local max) Randomize!!

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Summary

• Blind search
• Find some path to goal
• Depth-first, Breadth-first
• Iterative Deepening
• Analyzing search algorithms
• Completeness, Optimality, Time, Space
• Next time
• A little knowledge is a useful thing...

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Search Costs